Dom Dwyer scored his team-record 24th goal of the year across all competition, but Red Bulls forward Bradley Wright-Phillips scored his second goal of the game in the 90th minute to complete the comeback.
In the postseason for a fourth straight year and returning to the playoffs as the reigning MLS Cup champions, Sporting Kansas City fell to the Red Bulls for a third time in 2014 and suffered their second such defeat in a five-day span.
New York, who earned their first playoff victory at Red Bull Arena and snapped a postseason home winless streak that dated back to 2005, now advances to the Eastern Conference Semifinals against No. 1 seed D.C. United.
The two sides -- meeting for the first time in the playoffs in their 19-year history -- played to a scoreless first half that saw scoring chances on both ends of the field. Australian international Tim Cahill sent his shot over the crossbar from six yards out in the 17th minute, as did Wright-Phillips with a header 15 minutes later.
Thierry Henry found space on the left side of the penalty area in the 34th minute only to push his shot wide of the far post. Sporting Kansas City responded less than a minute later as Dwyer chased down a long ball from Aurelien Collin and picked out Paulo Nagamura, whose first-time shot was saved by Luis Robles.
The second half started emphatically with New York nearly finding the breakthrough goal in the 48th minute. Henry's free kick into the area was punched away by Eric Kronberg and fell to Dax McCarty at the top of the 18-yard box. The eight-year MLS veteran directed his header narrowly wide of the empty net.
Instead it was Sporting Kansas City that recorded the game's opening goal in the 53rd minute. Benny Feilhaber forced a turnover at midfield and ran at goal before slipping the ball centrally to Dwyer. The Englishman made no mistake, burying a first-time shot with his left foot past Robles for his second career postseason goal.
The Red Bulls rally would commence immediately as the East's top scoring team forced three saves in a four-minute span. Kronberg made his best save of the night in the 55th minute to tip an an Henry header over the crossbar, followed by a save on Jamison Olave's header on the ensuing corner kick and finally a stop on Lloyd Sam with two New York players looming.
The equalizer came in the 77th minute courtesy of Wright-Phillips, the MLS Golden Boot winner who matched the MLS single-season record with 27 goals in the regular season. Henry put the ball into his path and his right-footed shot took a deflection off Matt Besler before hitting the back of the net to set the stage for the last-minute dramatics.
In the final minute of regulation, Wright-Phillips rose to meet a cross from second-half substitute Ambroise Oyongo and directed his header off the inside of the far post for the game-winning goal. Wright-Phillips goal was his sixth against Sporting KC in 2014, including braces in back-to-back games over the past week.
The loss snapped Sporting Kansas City's 45-game unbeaten streak (43-0-2) in MLS competition (including playoffs) when leading in the 75th minute dating back to 2011 and brings the 2014 campaign to a close after 43 competitive matches.
With Thursday's elimination, Sporting Kansas City will hold the 11th position in Major League Soccer's various player acquisition mechanisms (i.e. MLS SuperDraft, allocation order, etc.) for the upcoming 2015 season.
